Choose an application of engineering where differential equations are used.
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matricesfinal2014_1.pptLAST TOPICS
- Homogeneous linear equations
- Eigenvalues and eigenvectors
Eigenvalues and Eigenvectors
An n×n matrix A multiplied by n×1 vector v results in another n×1 vector y=Av. Thus A can be considered as a transformation matrix.
In general, a matrix acts on a vector by changing both its magnitude and its direction. However, a matrix may act on certain vectors by changing only their magnitude, and leaving their direction unchanged (or possibly reversing it). These vectors are the eigenvectors of the matrix.
A matrix acts on an eigenvector by multiplying its magnitude by a factor, which is positive if its direction is unchanged and negative if its direction is reversed. This factor is the eigenvalue associated with that eigenvector.
Matrix expands or shrinks any vector lying in direction of eigenvector by scalar factor
Expansion or contraction factor is given by corresponding eigenvalue
Let x be an eigenvector of the matrix M. Then there must exist an eigenvalue λ such that Mx = λx or, equivalently,
Mx - λx = 0 or
(M – λI)x = 0
If we define a new matrix B = M – λI, then
Bx = 0
If B has an inverse then x = B-10 = 0. But an eigenvector cannot be zero.
Thus, it follows that x will be an eigenvector of M if and only if B does not have an inverse, or equivalently det(B)=0, or
det(M – λI) = 0
The roots of this equation determine the eigenvalues of M.
HOW TO FIND THEM
